APStatistics

Statistics Assignment 14: Inference about the Model

 

Directions: Complete the assignment on your own paper. Clearly label each answer. Your answers for this assignment must include reasons; simply stating the answer without justification will earn partial credit. (48 points)

 

A teacher asked her 8 introductory statistics students to record the total amount of time they spent studying for a particular test.  The amounts of study time x (in hours) and the resulting test grades y are given below.

    x   2   1    1.5   0.5   1    3    0    2

    y   92   81    84    68    85    96    48    74

 

1.   Make a scatterplot of the data. (2 points)

 

2.   Use your TI-83 to obtain the equation of the least-squares regression line and the correlation. (2 points)

 

3.   Explain in words what the slope b of the true regression line says about hours studied and grade awarded. (2 points)

 

4.   What is the estimate of b from the data?  What is your estimate of the intercept a  of the true regression line? (2 points)

 

5.   Use your calculator to calculate the residuals.  Report the sum of the residuals and the sum of the squares of the residuals.  Then use these results to estimate the standard deviation s in the regression model.  (3 points)

 

6.   The standard error of the slope SE_b  is defined as

            Calculate SE_b. (3 points)

 

7.   Suppose we want to find out if the number of hours studied helps predict grade awarded on this statistics test.  Formulate null and alternative hypotheses about the slope of the true regression line.  State a two-sided alternative. ( 2 points)

 

8.   Determine the test statistic, the degrees of freedom, and the P-value of t against the alternative.  (3 points)

 

9.   Would you reject the null hypothesis at the 1% significance level?  Explain briefly. (2 points)

 

10. Write your conclusions in plain language. (2 points)

 

11. Compute a 95% confidence interval for the slope ß of the true regression line. (2 points)

 

A mathematics professor wishes to analyze the relationship between the number of papers (in hundreds) graded by his department's student homework graders and the total amount of money paid to the graders.  He collects data for 12 randomly chosen graders and uses MINITAB to do regression analysis.  Below is a portion of the MINITAB output. (Here, COST = amount paid, PAPERS = # papers in hundreds, and the intervals listed at the bottom are computed for 1,600 papers.) 

The regression equation is

COST = 35.8 + 12.1 PAPERS

 

Predictor   Coef        Stdev       t-ratio     P

Constant    35.80       17.06       2.10       0.062

PAPERS      12.0835     0.9738      12.41      0.000

 

s = 6.526         R-sq = 93.9%            R-sq (adj) = 93.3%

 

Fit     Stdev. Fit             95% C.I.                95% P.I.

229.13        2.34       (223.93 , 234.34)       (213.68 , 244.58)

 

 

12. Formulate null and alternative hypotheses about the slope of the true regression line.  Adopt the two-sided alternative. (2 points)

 

13. What is the least-squares regression equation? (2 points)

 

14. What is the standard error about the line (also known as the standard deviation s in the regression model)? (2 points)

 

15. What is the slope of the least-squares regression line?  (2 points)

 

16. The model for regression inference has three parameters: a, b, and s.  Estimate these parameters from the data. ( 3 points)

 

17. What is the value of the test statistic for testing the hypotheses?  (2 points)

 

18. How many degrees of freedom does t have? ( 2 points)

 

19. What is the P-value for the test?   (2 points)

 

20. Is the number of papers graded useful for predicting the amount paid?  Use a significance level of 0.01.  Explain briefly. ( 2 points)

 

21. What is the estimated cost of grading 1,600 papers?  ( 2 points)

 

 

22. Find the 95% confidence interval for the average amount paid to all graders who grade 1,600 papers. ( 2 points)

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